The bit-wise decoder (B-DEC) in Fig. 3.4 operates on the L-values provided by the demapper Φ−1. The demapper acts independently of the B-DEC and calculates a vec- tor of L-values L using the max-log approximation. For later use, we represent the vector of L-values as a concatenation of length-m vectors, i.e., L = [L1, . . . , LN], where
Lk= [L1,k, . . . , Lm,k] are the L-values obtained from the kth observation. We consider the
max-log L-values in (2.20) with an additional scaling σ2(2d)−1for notational convenience, i.e., Lk,j , 1 4d " min s∈Sj,0 (Yk− s)2− min s∈Sj,1 (Yk− s)2 # . (3.24)
The scaling is irrelevant for the B-DEC though it may be important if other decoders are used, e.g., a decoder based on the sum product algorithm [59, Ch. 5]. The calculated L-values are passed to the B-DEC that implements the decision rule (2.18), which can be rewritten using the introduced notation as
ˆ CB , Φ−1B arg max u∈B {ul ⊺} ! . (3.25)
For the transmitted codeword b, the probability of detecting another codeword ˆb is PEPB(b, ˆb) = Pr{(2b − 1)L⊺− (2ˆb − 1)L⊺<0} = Pr{∆B(b, ˆb) < 0}, (3.26) where
∆B(b, ˆb) , (b − ˆb)L⊺ (3.27)
can be interpreted as a distance between the codewords when the B-DEC is used. In [35, Ch. 4], −∆B(b, ˆb) was called a pairwise score and the distribution of the pairwise score was analyzed under different assumptions. In this chapter, we are also interested in the distribution of ∆B(b, ˆb). However, our main goal is to compare this distribution with that of ∆X(x, ˆx) in (3.20). Since the mapping between b and x is one-to-one, with a slight abuse of notation, we use ∆B(x, ˆx) instead to highlight the similarity with (3.20).
3.3 Symbol-Wise Decoder Versus Bit-Wise Decoder 29
Table 3.2: SMDs as different combinations of L-values conditioned on different transmitted symbols for the BRGC.
ˆxi s1(000) s2(001) s3(011) s4(010) s5(110) s6(111) s7(101) s8(100) xi s1(000) 0 −L3 −L2− L3 −L2 −L1− L2 −L1− L2− L3 −L1− L3 −L1 s2(001) L3 0 −L2 −L2+ L3 −L1− L2+ L3 −L1− L2 −L1 −L1+ L3 s3(011) L2+ L3 L2 0 L3 −L1+ L3 −L1 −L1+ L2 −L1+ L2+ L3 s4(010) L2 L2− L3 −L3 0 −L1 −L1− L3 −L1+ L2− L3 −L1+ L2 s5(110) L1+ L2 L1+ L2− L3 L1− L3 L1 0 −L3 L2− L3 L2 s6(111) L1+ L2+ L3 L1+ L2 L1 L1+ L3 L3 0 L2 L2+ L3 s7(101) L1+ L3 L1 L1− L2 L1− L2+ L3 −L2+ L3 −L2 0 L3 s8(100) L1 L1− L3 L1− L2− L3 L1− L2 L2 −L2− L3 −L3 0 We can rewrite (3.27) as ∆B(x, ˆx) = XN k=1 ΛB(x k,ˆxk), (3.28) where ΛB(x k,ˆxk) , (Φ−1S (xk) − Φ−1S (ˆxk))L⊺k (3.29)
is the SMD4 for the B-DEC. Since the channel is memoryless, the SMDs for different
k = 1, . . . , N are independent. Once the distributions of all SMDs are known, the PDF of ∆B(x, ˆx) in (3.28) can be found as a convolution of the PDFs of the summands. The PEP in (3.26) is then given by the integral of the negative tail of this PDF. In the following, we discuss distributions of SMDs for a given time instant k and we omit k for clarity.
In order to proceed further, we need to specify the labeling used by the demapper since it greatly affects the SMD in (3.29). We concentrate on the 8-PAM constellation labeled with the BRGC. Different SMDs ΛB(x, ˆx) can then be expressed in terms of the L-values Lj, j = 1, 2, 3 as shown in Table 3.2. The binary labels for the constellation
symbols are also shown so that the entries of the table become evident.
The performance of the B-DEC is highly dependent on the distributions of the L- values. The distribution of the L-values in a certain bit position may depend on whether zero or one was transmitted. This complicates the analysis and some simplifications are usually made. In [43], a time-varying labeling was proposed in order to symmetrize the channel. The same effect can be achieved by using a random scrambler as in [35] or [68]. This makes the SMDs for the pairs of symbols whose labels differ in the same bits to have the same distribution. For example, the SMDs on the main anti-diagonal ΛB(s
1, s8), ΛB(s2, s7), . . . , ΛB(s8, s1) in Table 3.2 should all have the same distribution under such an assumption. This, however, is not generally true. The distributions of the aforementioned SMDs are given by the distribution of L1 conditioned on different transmitted symbols, and hence, are different. The assumption of a random scrambler gives good predictions
4−ΛB
30 Practical Approaches to Coded Modulation −8 −6 −4 −2 0 2 4 6 8 −6 −4 −2 0 2 4 6 y/d l2 /d s1 s2 s3 s4 s5 s6 s7 s8
Figure 3.5: Approximation of L-values. The solid line shows (normalized) l2 as a function of the (normalized) observation y. The dashed line shows the approximation (consistent and ZC) of the L-value. The dash-dotted line shows the distribution fY |X(y|s6) for d/σ = −5 dB; the distribution is scaled for illustration purposes. Empty and filled circles show constellation points labeled with 0 and 1, respectively.
of the PEP but cannot be used when the S-DEC and the B-DEC are compared. Here, we do not use this assumption and treat every SMD in Table 3.2 individually.
Distribution of L-values
As mentioned in Section 2.2.4, the max-log L-values are linear functions of the observation. This implies that the SMDs in Table 3.2 are linear functions of the observation as well. In this section, we show how to obtain the PDFs of two SMDs in Table 3.2 and how these PDFs can be approximated. The distributions of all other SMDs in Table 3.2 can be obtained and approximated in a similar fashion.
It follows from Table 3.2 that the SMD ΛB(s6, s7) = L2 and hence, its distribution is the distribution of L2 given that symbol s6 was transmitted. Fig. 3.5 shows the piecewise linear function l2(y) together with the distribution of the observation conditioned on the transmitted symbol s6 for d/σ = −5 dB. The PDF fL2|X(l|s6) is thus a sum of
piecewise Gaussian functions with mean and variance determined by the parameters of the corresponding linear pieces of l2(y). The PDF fL2|X(l|s6), or equivalently, the PDF
of ΛB(s6, s7), is shown in Fig. 3.6 with the solid line.
The PDFs of the SMDs are easy to obtain but they become analytically intractable when several PDFs need to be convolved, and hence, approximations of the distributions are usually used. We consider two ways to approximate such PDFs. The first one is the so-called consistent approximation [66] for which the L-value is approximated by a linear function corresponding to the linear piece over the Voronoi region of the transmitted symbol. This approximation is shown with the dashed line in Fig. 3.5. The approximated PDF is thus a single Gaussian function that approximates the exact PDF “at the mean” of the L-value and it is shown with the dashed line in Fig. 3.6. The second approach approximates the negative tail of the PDF, which is more important when analyzing
3.3 Symbol-Wise Decoder Versus Bit-Wise Decoder 31 −80 −6 −4 −2 0 2 4 6 8 0.05 0.1 0.15 0.2 0.25 λ/d fΛ B ( s6 ,s7 ) (λ )d
Figure 3.6: Distribution of the SMD λB(s
6, s7) for d/σ = −5 dB. The solid line represents the exact PDF and the dashed line shows the approximated PDF. The consistent and the ZC approximations give the same results.
the PEP. For that, the L-value is approximated using the so-called zero-crossing (ZC) approximation [69], i.e., the L-value is approximated by a linear function corresponding to the linear piece of the L-value function intersecting the zero-level at the closest point to the transmitted symbol. For the example in Fig. 3.5, the ZC approximation coincides with the consistent approximation (dashed line) and gives the same approximated PDF, which is shown with the dashed line in Fig. 3.6. However, the two approximations are not always the same. In the following, we give an example showing that the ZC approximation should be preferred when analyzing the probability of error.
As shown in Table 3.2, the SMD ΛB(s6, s8) = L2 + L3. It is shown in Fig. 3.7 as a piece-wise linear function of the observation y together with the conditional distribution of the observation fY |X(y|s6). The exact PDF of ΛB(s6, s8) is a sum of piece-wise Gaussian functions shown in Fig. 3.8. Due to the horizontal parts of the SMD in Fig. 3.7, the exact PDF contains a Dirac delta function with amplitude A = 1−2Q(d/σ)+Q(5d/σ)−Q(7d/σ) at λ = 2d. The consistent approximation approximates the SMD by a linear function
λ= 2d and hence, the approximated PDF is a Dirac delta function of unit amplitude at
λ= 2d. Clearly, such an approximation gives poor results when the PEP is analyzed. The ZC approximation of the SMD ΛB(s6, s8) is shown with the dashed line in Fig. 3.7 and it results in a Gaussian distribution depicted with the dashed line in Fig. 3.8. It is possible to show that the ZC approximation is asymptotically tight (when γ → ∞) in terms of PEP, if ∆B(x, ˆx) in (3.28) consists of only one SMD. We also considered different ∆B(x, ˆx) consisting of several SMDs, for instance ∆B(x, ˆx) for the codewords in [Paper B, Fig. 2], and the asymptotic tightness of the ZC approximation was successfully verified. For a general case, where ∆B(x, ˆx) is a summation of an arbitrary number of different SMDs, the tightness of the ZC approximation remains an open question.
Once the parameters of the approximated distributions (the mean and the variance) for all entries of Table 3.2 are known, they can be compared to those of the S-DEC. We performed this analysis for 4-PAM labeled with a Gray code in [Paper B]. The results showed that the distribution parameters are very similar for the two decoders, which
32 Practical Approaches to Coded Modulation −8 −6 −4 −2 0 2 4 6 8 −6 −4 −2 0 2 4 6 y/d λ B(s 6 ,s8 )/ d s1 s2 s3 s4 s5 s6 s7 s8
Figure 3.7: Approximation of the SMD. The solid line shows the (normalized) SMD
λB(s6, s8) as a function of the (normalized) observation y. The dashed line shows the ZC approximation of the SMD. The dash-dotted line shows the distribution fY |X(y|s6) for
d/σ= −5 dB; the distribution is scaled for illustration purposes.
−80 −6 −4 −2 0 2 4 6 8 0.05 0.1 0.15 0.2 0.25 0.3 λ/d fΛ B ( s6 ,s8 ) (λ )d Ad
Figure 3.8: Distribution of the SMD λB(s6, s
8) for d/σ = −5 dB. The solid line rep- resents the exact PDF and the dashed line shows the approximated PDF using the ZC approximation. The PDF obtained based on the consistent approximation is a Dirac delta function with amplitude 1 at λ = 2d and is not shown in the figure.
allowed us to bound the asymptotic performance loss of the B-DEC compared to the S-DEC. The analysis also showed that for a wide range of codes, the performance loss is equal to zero. This phenomenon is illustrated in Fig. 3.2 for 8-PAM. The dash-dotted line marked circles shows the performance of the B-DEC with the BRGC for the TCM scheme (CBRGC, GBRGC). The B-DEC causes a loss of fractions of a dB compared to the S-DEC shown with the solid line marked with circles.
A similar analysis could be performed for other than Gray labelings. For example, we derived the distribution parameters of the SMDs for 4-PAM labeled with the NBC. Unfortunately, the results do not allow us to bound the performance loss and draw quan- titative conclusions. However, they clearly indicate that the loss can be large when the B-DEC is used with the NBC. We illustrate this in Fig. 3.2 for 8-PAM. The dotted line marked with circles shows the performance of the B-DEC with the NBC for the TCM
3.3 Symbol-Wise Decoder Versus Bit-Wise Decoder 33
scheme (CBRGC, GBRGC). For that, the demapper for the NBC together with a decoder for the binary code CNBC is used at the receiver. As we can see, the loss is approximately 4.5 dB compared to the S-DEC. This gives an intuition on why the B-DEC performs bad when used with a non-Gray labeling and motivates the use of iterative decoders to obtain a near-optimal performance.